If a, b, c are in AP, x is the GM between a and b; y is the GM between b and c; then show that b² is the AM between x² and y².

To prove: b² is the AM between x² and y².

Given: (i) a, b, c are in AP

(ii) x is the GM between a and b

(iii) y is the GM between b and c

Formula used: (i) Arithmetic mean between

(ii) Geometric mean between

As a, b, c are in A.P.

⇒ 2b = a + c … (i)

As x is the GM between a and b

⇒ x =

⇒ x² = ab … (ii)

As y is the GM between b and c

⇒ y =

⇒ y² = bc … (iii)

Arithmetic mean of x² and y² is

Substituting the value from (ii) and (iii)

Substituting the value from eqn. (i)

= b²

Hence Proved